Homework 6

5/6/02

Physics 110B

Griffiths 9.28, 9.30, 11.3

1. Two extremely large horizontal metal planes are separated by a distance $d$. They are highly conducting so that the electromagnetic fields inside the metal are zero. What is the minimum frequency that can propagate in the region of air between the planes?

2. A short microwave pulse of frequency $\omega$ is sent down a very long rectangular waveguide of length $L$ with width $a$ and height $b$. The pulse has not been specially prepared and so has both TE and TM components, furthermore one expects that all possible propagating modes will be excited. For the $mn$th mode to propagate, $\omega_{mn} < \omega$. Let us say that with the particular choice of height, width and $\omega$ this gives $N$ propagating modes. Describe what is received at the other end of the wave guide, in particular

(a) How many pulses will appear at the other end.

(b) What is the time of arrival of each pulse. Give your answer in terms of $\omega$ and $\omega_{mn}$.

Remember that $k_x^2 + k_y^2 + k_z^2 ~=~ (\omega /c)^2$ and that $k_y ~=~ m \pi / a$ and $k_z ~=~ n \pi / b$. Assume that there is no absorption at the walls of the waveguide.

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